There isn't much to do about this, but typically we have seen better results with higher quality KVM devices.
After initial connection, wireless keyboard response may take 10-15 sec. The KVMs USB specification must be USB 2.0 or greater USB 1.0 doesn’t have enough bandwidth to support 2 wireless HID devices from 1 port.
BEST KVM SWITCH KEYBOARD SHORTCUT DRIVER
In order to support Hot-Key operation and uninterrupted keyboard-mouse function while switching, most KVM Switches have a chip that constantly buffers (reads) signals from the keyboard and mouse, and transfers any button presses to the connected computer. Connecting Wireless Keyboard / Mouse to a KVM Switch